Name for vector without sign/sense? (Just module and direction) If vectors are the equivalence class of oriented segments, what is the name for the equivalence class of unoriented segments?
These "unsigned vectors" would have a module (or normal value) and a direction but no sense/sign.
An example of a use for such entities is the cross product: if we don't know the orientation/headiness of the space, we can "unify" the two possible answers as a single "unsigned vector".
 A: I would simply call this a line, which in a sense is a vector that points in both directions, and only unsigned direction matters. Whereas a vector space is a set of vectors, the set of all lines in a vector space is the corresponding projective space.
If you want to preserve magnitude as well, then it's a bit less clear-cut what to call it. You could say it's a line segment, though that is a more general term. Even a line segment through the origin could fail to have a well-defined magnitude. The magnitude could be the length, but you would really want it to be symmetric so that the length is the same in both directions. We are really looking at vectors that are identified by the equivalence relation $v\sim -v$, and these equivalence classes are the objects you seek. They can be represented by one or the other vector, but I wouldn't say they have a specific name.
A: As pointed out by @Matt Samuel, I would use the following definition for the line segment $l$ between two points $A$ and $B$:
$$l := \left\{ tA + (1-t)B \mid t \in [0, 1] \right\}$$
Hope that helps, please let me know if I can help further !
A: I do not know of any special name for this concept.
In the context of Graph theory
there is a clear distinction between directed and
undirected graphs. The edges are sometimes known as
"lines". Thus, in graph theory, both undirected lines
and directed lines appear on an equal footing. Then,
depending on context, a "line" is assumed to be either directed or undirected.
In the context of projective space the
fundamental objects are "points" which are defined
to be equivalence classes of vectors that are said
to be related if they are nonzero scalar multiples
of each other. In your case the relation is much
more restricted and only a vector and its negative
are related.
Better is the context of convex sets in Euclidean
space. Any two distinct points determine the set of
convex linear combinations of the two points. This
is the line segment with the two given points as the
end points of the segment. The displacement from one
of the endpoints to the other is a vector in the
vector space associated to the Euclidean space. Of
course, the displacement going the other way is the
negative of the first vector.
Despite this, I don't know of any special name for
the equivalence class of a vector and its negative. However, in the specific case
that you mention of the cross product of
two vectors, there is the concept of
pseudovector. The Wikipedia article states:

In three dimensions, a pseudovector is associated with the curl of a polar vector or with the cross product of two polar vectors

It also states:

A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics, pseudovectors are equivalent to three-dimensional bivectors, from which the transformation rules of pseudovectors can be derived.

Thus, the reason for confusion about the
status of the cross product is that the
result is better thought of as a bivector
and not as an actual ordinary or polar
vector.
